How Do Laurent Series Differ from Taylor Series in Complex Analysis?

Click For Summary
SUMMARY

The discussion clarifies the distinctions between Laurent Series (LS) and Taylor Series (TS) in complex analysis. LS can be defined for functions with isolated singularities, while TS is applicable only to holomorphic functions around a point. It is established that all TS are LS, but not all LS are TS. Furthermore, TS is limited to regions within a disk radius, whereas LS can be defined for annuli, allowing for greater flexibility in function expansion.

PREREQUISITES
  • Understanding of complex analysis concepts, particularly holomorphic functions.
  • Familiarity with the definitions and properties of Laurent Series and Taylor Series.
  • Knowledge of singularities in complex functions.
  • Basic grasp of analytic functions and their expansions.
NEXT STEPS
  • Study the properties of Laurent Series in detail, focusing on their application to functions with singularities.
  • Learn about the conditions under which a function can be expressed as a Taylor Series.
  • Explore examples of functions that exhibit both TS and LS to understand their practical applications.
  • Investigate the concept of annuli in complex analysis and how they relate to function expansion.
USEFUL FOR

Students and professionals in mathematics, particularly those specializing in complex analysis, as well as educators looking to clarify the differences between series expansions in their teaching materials.

ognik
Messages
626
Reaction score
2
My book is a little confusing sometimes, and googling doesn't always help. Just a couple of queries - and please add any of your own 'tips & tricks'...

1) Laurent series (LS) is defined from $ -\infty $, yet all the examples I have seen start from 0 - I can't think of an annulus with a negative radius myself, so do I just use it from 0 and not worry about the negative side of the domain?

2) What is the practical difference between a complex Taylor series (TS) and LS? I have seen suggested that TS is for holomorhpic functions and LS for isolated singularities, but it seems to me those conditions could apply to both TS & LS?

3) A difference I can see is that TS only allows for the region < disk radius, but LS provide for > some R (and also within an annulus) - so for an annulus could we use TS for inside the large radius, LS for outside the smaller radius?

4) For an annulus, couldn't we avoid using LS, Juts take TS of the outer - TS of the inner?

5) Conversely, could we use LS instead of TS, by making the smaller radius 0?
Thanks for all advice.
 
Physics news on Phys.org
ognik said:
My book is a little confusing sometimes, and googling doesn't always help. Just a couple of queries - and please add any of your own 'tips & tricks'...

1) Laurent series (LS) is defined from $ -\infty $, yet all the examples I have seen start from 0 - I can't think of an annulus with a negative radius myself, so do I just use it from 0 and not worry about the negative side of the domain?

There is no relation between the negative radius and the index of the summation. Having a negative sign in the summation means that the function is not holomorphic at least in a circle of some radious.

2) What is the practical difference between a complex Taylor series (TS) and LS? I have seen suggested that TS is for holomorhpic functions and LS for isolated singularities, but it seems to me those conditions could apply to both TS & LS?

All TS are LS but not vice versa. As you said they are used to expand functions with singularities.

3) A difference I can see is that TS only allows for the region < disk radius, but LS provide for > some R (and also within an annulus) - so for an annulus could we use TS for inside the large radius, LS for outside the smaller radius?

By definition of TS it is used to expand the function around a point and the function has to be analytic on that point. So it has no meaning to say that the function has a TS on an annulus.

4) For an annulus, couldn't we avoid using LS, Juts take TS of the outer - TS of the inner?

If the function is holomorphic on an annulus then it has no TS in the inner radius.

5) Conversely, could we use LS instead of TS, by making the smaller radius 0?
Thanks for all advice.

Yes that's way I said that every TS is LS.
 

Similar threads

Graduate Laurent series for algebraic functions
  • · Replies 9 ·
Replies
9
Views
3K
MHB Understanding the Laurent Series of $\frac{1}{z(z-1)(z-2)}$
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad About convergence of complex power series on the circle ##|z - z_0|=r##
  • · Replies 22 ·
Replies
22
Views
4K
Graduate Can someone explain the need for Laurent series in complex analysis?
  • · Replies 10 ·
Replies
10
Views
5K
MHB Find coefficiant of Laurent series, without using residue
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Laurent series - where does f(z) need to be holomorphic?
  • · Replies 2 ·
Replies
2
Views
2K
MHB A guide to Calculating the Laurent Series of $\frac{33}{(2z - 1)(z + 5)}$
  • · Replies 2 ·
Replies
2
Views
2K
Laurent Series (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Help understanding Laurent series in complex analysis
  • · Replies 8 ·
Replies
8
Views
16K
Graduate Radius of Convergence: Complex Series Need Not Be Defined Everywhere
  • · Replies 3 ·
Replies
3
Views
3K